13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- disappear if the number of atoms in it 2 g (the complete evaporation). Number of atoms in each bubble is controlled by the next equation: 2 1 1/3 4 2 exp ln 1 3 v c c P R dg R g dt c P kT g , (16) where 1/3 1 1 3 / 4 v R m . (17) Pressure and temperature of the caring agent phase and of the dispersed phase have been obtained from the metastable equation of state: , , , , c d c d c d P P U , , , , , c d c d c d T T U . This equation of state has been used for determination of the density v and pressure of the saturated vapor as well. Than the bubbles became large enough they merge and form a simply connected phase. On the contrary, the liquid divides on separate drops. These drops are the condensation centers in the expanding and cooling vapor. Number and sizes of the liquid drops are determined by the number and sizes of the vapor bubbles at the moment of merging. We have established this connection in a simplest case of equal distance between the bubbles centers. Bubbles merge then their diameters 2R reach the value of a distance between their centers. Single bubble occupy the volume 3 4 /3 R , liquid volume per one bubble at this time is 3 8 4 /3 R . We have assumed that the number of resulting drops is equal to the number of vapor bubbles, and then 3 8 4 /3 R is the volume of one liquid drop. Due to the surface tension, the drops obtain a spherical form after the liquid phase division, and the radius of such spherical liquid drop can be estimated as 1/3 L 6/ 1 R R R . It follows, that the merging passes then the vapor volume fracture achieves the value 0.5 c . A number of smaller drops can be formed during the vapor bubbles merging, but we neglect it here. 3. Numerical investigation of the metal fracture We have numerically investigated the copper irradiation by the high-current pulsed electron beam with parameters: the energy of fast electrons is 1 MeV, the beam current density is 10 kA/cm2, the pulse duration is 50 ns. The beam action causes the sharp heating of the substance (up to 4900 K) in the energy release zone (Fig. 1) and the formation of area of the high-pressure - up to 17 GPa (Fig. 2). The substance temperature exceeds the melting temperature up to the depth of 0.4 mm; in this layer the metal is melted. Release of the high pressure area results in formation of the compression wave with the amplitude up to 11.5 GPa. The compression wave front is becoming sharper with the time, and it transforms into the shock wave. Reflection from the free (irradiated) surface forms the rarefaction wave, following behind the shock wave. This rarefaction wave creates in the liquid metal the negative stress with the value up to 2.5 GPa, which initiates the fracture of the liquid phase through the generation and growth of the vapor bubbles. The generation and growth of the bubbles result in reduction of the liquid metal volume and, therefore, it releases the tensile stresses; the substance passes in the equilibrium two-phase state. This process restricts the tensile stresses; otherwise, the rarefaction wave amplitude would be the same as the amplitude of the compression wave. It should be noted that the existence of metastable liquid state leads to propagation of the rarefaction (tensile) wave inside the metal behind the shock wave. Negative stresses in the rarefaction wave can reach 2.5 GPa. Thus, the structure of stresses in the metal differs from that in calculations [4,8],
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